Close the selection lock caused by mixture-based atom scoring

Develop and validate a joint selection criterion for the vocals and accompaniment atom spans that explains the mixture while penalising overlap between the spans, thereby eliminating the cross-source term that causes atoms to be selected for explaining the mixture rather than resembling their respective sources.

Background

The rigid gain-and-delay class repairs the degeneracy of the free per-bin deformation by making the selection residual informative, but it does not solve the remaining selection error. Candidate atoms are scored against the mixture rather than against their own source, so the score includes a cross-term measuring how well an atom from one source explains the other source.

The paper proposes a joint criterion that would select atom spans for explaining the mixture while penalising overlap between the two source spans, for example through principal-angle penalties or a penalty based on the cross-Gram matrix. Two approximations improve the results but recover only 8–13% of the remaining selection lock, leaving the problem unresolved. Because the reconstruction capacity is already present in the candidate atoms, closing this lock could directly improve the delivered separation quality.

References

The lightest remedy, and the closest to the diagnosis, is a joint criterion over both sources: choose $(\mathbf{A}_1, \mathbf{A}_2)$ to explain $\mathbf{x}$ while penalising configurations in which the two spans overlap, that overlap being the mechanism by which explaining the mixture twice beats resembling either source once. A penalty on the principal angles between $\spn \mathbf{A}_1$ and $\spn \mathbf{A}_2$, or equivalently on $|\mathbf{A}_1{*}\mathbf{A}_2|$, expresses that directly and leaves the inference structure untouched. Two approximations of it were measured on the protocol below, one discounting each atom's score by its coherence with the other source's dictionary and one selecting both sets jointly by deflation: at $k = 16$ and 300 atoms they raise the vocals from $+5.57$ to $+6.20$ and $+6.54$~dB against a capacity of $+12.99$, which closes 8 to 13 per cent of the lock and leaves it open.

— What Selects, What Reconstructs: Repairing Exemplar-Based Complex-Spectrum Separation  (2609.04756 - Baelde, 4 Sep 2026) in Section 3, subsection “Selection Against the Mixture” (label sec:lock)